How does BSM connect to the law of one price and no-arbitrage?

The law of one price says: two portfolios with identical payoffs in every possible future state must trade at identical prices today. Otherwise an arbitrageur sells the expensive one, buys the cheap one, and pockets the difference with zero risk.

Applied to BSM:

The replicating portfolio ($\Delta$ shares + $B$ in cash, rebalanced continuously) matches the call payoff $\max(S_T - K, 0)$ in every state. By the law of one price, the call must therefore trade at exactly the cost of the replicating portfolio.

What happens if BSM is "violated":

Suppose the market call price is $12 but the replicating portfolio costs only $10. Here is the arbitrage:

The arbitrageur is short the call (a liability that pays $\max(S_T - K, 0)$) and long the replicating portfolio (an asset that pays $\max(S_T - K, 0)$). The positions cancel at $T$ and the $2 + interest is locked in at $t=0$.

In real markets:

BSM "violations" happen all the time, but they reveal themselves as rich implied vol or transaction cost rather than free money. Real arbitrage requires:

• No transaction costs
• No bid-ask spread
• Continuous, perfectly liquid rebalancing
• Stable, known vol

When those conditions break (which is always), the no-arb bound widens into a band. Inside the band, the option can trade away from theoretical BSM without an arbitrage opportunity. This is why traders watch implied vol — it is the slack between market price and theoretical BSM, calibrated for whatever assumptions are violated.

For the exam: the law of one price + replication = BSM is "exactly right." For trading, "BSM with the right vol surface" is "approximately right." Hold both ideas in your head.